# Coherent Detection of Turbo-Coded OFDM Signals Transmitted Through Frequency Selective Rayleigh Fading Channels with Receiver Diversity and Increased Throughput

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## Abstract

In this work, we discuss techniques for coherently detecting turbo coded orthogonal frequency division multiplexed (OFDM) signals, transmitted through frequency selective Rayleigh (the magnitude of each channel tap is Rayleigh distributed) fading channels having a uniform power delay profile. The channel output is further distorted by a carrier frequency and phase offset, besides additive white Gaussian noise. A new frame structure for OFDM, consisting of a known preamble, cyclic prefix, data and known postamble is proposed, which has a higher throughput compared to the earlier work. A robust turbo decoder is proposed, which functions effectively over a wide range of signal-to-noise ratio (SNR). Simulation results show that it is possible to achieve a bit-error-rate (BER) of \(10^{-5}\) at an SNR per bit as low as 8 dB and throughput of 82.84 %, using a single transmit and two receive antennas. We also demonstrate that the practical coherent receiver requires just about 1 dB more power compared to that of an ideal coherent receiver, to attain a BER of \(10^{-5}\). The key contribution to the good performance of the practical coherent receiver is due to the use of a long preamble (512 QPSK symbols), which is perhaps not specified in any of the current wireless communication standards. We have also shown from computer simulations that, it is possible to obtain even better BER performance, using a better code. A simple and approximate Cramér–Rao bound on the variance of the frequency offset estimation error for coherent detection, is derived. The proposed algorithms are well suited for implementation on a DSP-platform.

## Keywords

OFDM Coherent detection Matched filtering Turbo codes Frequency selective Rayleigh fading Channel capacity## 1 Introduction

- 1.
The Myth: Sixty years of research following Shannon’s pioneering paper has led to telecommunications solutions operating arbitrarily close to the channel capacity—“flawless telepresence” with zero error is available to anyone, anywhere, anytime across the globe.

- 2.
The Reality: Once we leave home or the office, even top of the range iPhones and tablet computers fail to maintain “flawless telepresence” quality. They also fail to approach the theoretical performance predictions. The 1,000-fold throughput increase of the best third- generation (3G) phones over second-generation (2G) GSM phones and the 1,000-fold increased teletraffic predictions of the next decade require substantial further bandwidth expansion toward ever increasing carrier frequencies, expanding beyond the radio- frequency (RF) band to optical frequencies, where substantial bandwidths are available.

- 1.
maximize the bit-rate

- 2.
minimize the bit-error-rate

- 3.
minimize transmit power

- 4.
minimize transmission bandwidth

- 1.
was signal processing for coherent communications given a chance to prove itself, or was it ignored straightaway, due to “complexity” reasons?

- 2.
are the present day single antenna wireless transceivers, let alone multi-antenna systems, performing anywhere near channel capacity?

In this article, we dwell on coherent receivers based on orthogonal frequency division multiplexing (OFDM), since it has the ability to mitigate intersymbol interference (ISI) introduced by the frequency selective fading channel [6, 7, 8]. The “complexity” of coherent detection can be overcome by means of parallel processing, for which there is a large scope. We wish to emphasize that this article presents a proof-of-concept, and is hence not constrained by the existing standards in wireless communication. We begin by first outlining the tasks of a coherent receiver. Next, we scan the literature on each of these tasks to find out the state-of-the-art, and finally end this section with our contributions.

- 1.
To correctly identify the start of the (OFDM) frame (SoF), such that the probability of false alarm (detecting an OFDM frame when it is not present) or equivalently the probability of erasure/miss (not detecting the OFDM frame when it is present) is minimized. We refer to this step as timing synchronization.

- 2.
To estimate and compensate the carrier frequency offset (CFO), since OFDM is known to be sensitive to CFO. This task is referred to as carrier synchronization.

- 3.
To estimate the channel impulse/frequency response.

- 4.
To perform (coherent) turbo decoding and recover the data.

A robust timing and frequency synchronization for OFDM signals transmitted through frequency selective additive white Gaussian noise (AWGN) channels is presented in [9]. Timing synchronization in OFDM is addressed in [10, 11, 12, 13, 14]. Various methods of carrier frequency synchronization for OFDM are given in [15, 16, 17, 18, 19, 20, 21]. Joint timing and CFO estimation is discussed in [22, 23, 24, 25, 26, 27].

Decision directed coherent detection of OFDM in the presence of Rayleigh fading is treated in [28]. A factor graph approach to the iterative (coherent) detection of OFDM in the presence of CFO and phase noise is presented in [29]. OFDM detection in the presence of intercarrier interference (ICI) using block whitening is discussed in [30]. In [31], a turbo receiver is proposed for detecting OFDM signals in the presence of ICI and inter antenna interference.

Most flavors of the channel estimation techniques discussed in the literature are done in the frequency domain, using pilot symbols at regular intervals in the time/frequency grid [32, 33, 34, 35, 36]. Iterative joint channel estimation and multi-user detection for multi-antenna OFDM is discussed in [37]. Noncoherent detection of coded OFDM in the *absence of frequency offset* and assuming that the channel frequency response to be constant over a block of symbols, is considered in [38]. Expectation maximization (EM)-based joint channel estimation and exploitation of the diversity gain from IQ imbalances is addressed in [39].

Detection of OFDM signals, in the context of spectrum sensing for cognitive radio, is considered in [40, 41]. However, in both these papers, the probability of false alarm is quite high (5 %).

In [42], discrete cosine transform (DCT) based OFDM is studied in the presence of frequency offset and noise, and its performance is compared with the discrete Fourier transform (DFT) based OFDM. It is further shown in [42] that the performance of DFT-OFDM is as good as DCT-OFDM, for small frequency offsets.

A low-power OFDM implementation for wireless local area networks (WLANs) is addressed in [43]. OFDM is a suggested modulation technique for digital video broadcasting [44, 45]. It has also been proposed for optical communications [46].

The novelty of this work lies in the use of a filter that is matched to the preamble, to acquire timing synchronization [47, 48] (start-of-frame (SoF) detection). Maximum likelihood (ML) channel estimation using the preamble is performed. This approach does not require any knowledge of the channel and noise statistics.

- 1.
It is shown that, for a sufficiently long preamble, the variance of the channel estimator proposed in eq. (28) of [4] approaches zero.

- 2.
A known postamble is used to accurately estimate the residual frequency offset for large data lengths, thereby increasing the throughput compared to [4, 5].

- 3.
Turbo codes are used to attain BER performance closer to channel capacity compared to any other earlier work in the open literature, for channels having a uniform power delay profile (to the best of the authors knowledge, there is no similar work on the topic of this paper, other than [4, 5]).

- 4.
A robust turbo decoder is proposed, which performs effectively over a wide range of SNR (0–30 dB).

This paper is organized as follows. Section 2 describes the system model. The enhanced frame structure is described in Sect. 3. The modifications in the turbo decoder in the presence of receive diversity and the variance of the channel estimation error, are presented in Sect. 4. The channel capacity is discussed in Sect. 5. The BER results from computer simulations are given in Sect. 6. Finally, in Sect. 7, we discuss the conclusions and future work.

## 2 System Model

## 3 Enhanced Frame Structure

Consider the frame in Fig. 1a. In addition to the preamble, prefix and data, it contains “buffer” (dummy) symbols of length \(B\) and postamble of length \(L_o\), all drawn from the QPSK constellation. In Fig. 1b we illustrate the processing of \(L_d\) symbols at the transmitter. Observe that only the data and postamble symbols are interleaved before the IFFT operation. After interleaving, the postamble gets randomly dispersed between the data symbols. The buffer symbols are sent directly to the IFFT, without interleaving. The preamble and the cyclic prefix continue to be processed according to Figure 1 in [4] and eq. (3) in [4]. We now explain the reason behind using this frame structure. In what follows, we assume that the SoF has been detected, fine frequency offset correction has been performed and the channel has been estimated.

- 1.
Modulation in the time domain results in a shift in the frequency domain. Therefore, any residual frequency offset after fine frequency offset correction, results in a frequency shift at the output of the FFT operation at the receiver. Moreover, due to the presence of a cyclic prefix, the frequency shift is circular. Therefore, without the buffer symbols, there is a possibility that the first data symbol would be circularly shifted to the last data symbol or vice versa. This explains the use of buffer symbols at both ends in Fig. 1. In order to compute the number of buffer symbols (\(B\)), we have to know the maximum residual frequency offset, after fine frequency offset correction. Referring to Fig. 3, we find that the maximum error in fine frequency offset estimation at 0 dB SNR per bit is about \(\pm 2\times 10^{-3}\) radians. With \(L_d=4{,}096\), the subcarrier spacing is \(2\pi /4{,}096=1.534\times 10^{-3}\) radians. Hence, the residual frequency error would result in a shift of \(\pm 2/1.534=\pm 1.3\) subcarrier spacings. Therefore, while \(B=2\) would suffice, we have taken \(B=4\), to be on the safe side.

- 2.
Since the frequency shift is not an integer multiple of the subcarrier spacing, we need to interpolate in between the subcarriers, to accurately estimate the shift. Interpolation can be achieved by zero-padding the data before the FFT operation. Thus we get a \(2L_d-\)point FFT corresponding to an interpolation factor of two and so on. Other methods of interpolation between subcarriers is discussed in [49].

- 3.After the FFT operation, postamble matched filtering has to be done, since the postamble and \(\hat{H}_k\approx \tilde{H}_k\) are known. The procedure for constructing the postamble matched filter is illustrated in Fig. 5. From simulations, it has been found that a postamble length \(L_o=128\) results in false peaks at the postamble matched filter output at 0 dB SNR per bit. Therefore we have taken \(L_o=256\). With these calculations, the length of the data works out as \(L_{d2}=L_d-2B-L_o=4{,}096-8-256=3{,}832\) QPSK symbols. The throughput of the enhanced frame structure (with rate-1 turbo code) isThe throughput comparison of various frame structures is summarized in Table 1.$$\begin{aligned} \fancyscript{T}&= \frac{L_{d2}}{L_p+L_{cp}+L_d} \nonumber \\&= \frac{3{,}832}{512+18+4{,}096} \nonumber \\&= 82.84\,\%. \end{aligned}$$(4)

Throughput comparison of various frame structures with \(L_p=L_{d1}=512\), \(L_{d2}=3{,}832\), \(L_{cp}=18\)

## 4 Receiver

The receiver algorithms for start-of-frame (SoF) detection, frequency offset, channel and noise variance estimation are already discussed in [4, 5], and apply also to the enhanced frame structure given above and receive diversity. In what follows, we describe the modifications required in the turbo decoder in the presence of receive diversity.

### 4.1 Turbo Decoding

### 4.2 Robust Turbo Decoding

If any element of \(\mathbf {b}_1'\) is less than say, \(-30\), then set it to \(-30\). Thus we get a normalized exponent vector \(\mathbf {b}_{1,\,\mathrm {norm}}\), whose elements lie in the range \([0,\, -30]\). It has been found from simulations that normalizing the exponents does not lead to any degradation in BER performance, on the contrary, it increases the operating SNR range of the turbo receiver. In practice, we could divide the range \([0,\, -30]\) into a large number (e.g. 3,000) of levels and the exponentials (\(\mathrm {e}^b\)) could be precomputed and stored in the DSP processor, and need not be computed in real-time. The choice of the minimum exponent (e.g. \(-30\)), would depend on the precision of the DSP processor or the computer.

### 4.3 Variance of the Channel Estimation Error

## 5 The Channel Capacity

- 1.
The sphere packing derivation of the channel capacity formula [50], does not require noise to be Gaussian. The only requirements are that the noise samples have to be independent, the signal and noise have to be independent, and both the signal and noise must have zero mean.

- 2.
The channel capacity depends only on the SNR.

- 3.
- 4.
It is customary to define the average SNR per bit (\(E_b/N_0\)) over two dimensions (complex signals). When the signal and noise statistics over both dimensions are identical, the average SNR per bit over two dimensions is identical to the average SNR per bit over one dimension. Therefore (19) is valid, even though the SNR is defined over one dimension and the SNR per bit is defined over two dimensions.

- 5.
The notation \(E_b/N_0\) is usually used for continuous-time, passband analog signals [50, 51, 52], whereas SNR per bit is used for discrete-time signals [7]. However, both definitions are equivalent. Note that passband signals are capable of carrying information over two dimensions, using sine and cosine carriers, inspite of the fact that passband signals are real-valued.

- 6.
Each dimension corresponds to a separate and independent path between the transmitter and receiver.

- 7.
The channel capacity is additive with respect to the number of dimensions. Thus, the total capacity over \(2N\) real dimensions is equal to the sum of the capacity over each real dimension.

- 8.
Each \(S_{k,\, 3,\, i}\) in (2) corresponds to one transmission (over two dimensions, since \(S_{k,\, 3,\, i}\) is complex-valued).

- 9.Transmission of \(L_{d2}\) data bits in Fig. 1 (for a rate-1 turbo code), results in \(NL_{d2}\) complex samples (\(2NL_{d2}\) real-valued samples) of \(\tilde{R}_{k,\, i,\, l}\) in (2), for \(N\)th-order receive diversity. Therefore, the channel capacity isper dimension. In other words, (20) implies that in each transmission, one data bit is transmitted over \(2N\) dimensions. Similarly, for a rate-\(1/2\) turbo code with \(N\)th-order receive diversity, transmission of \(L_{d2}/2\) data bits results in \(NL_{d2}\) complex samples of \(\tilde{R}_{k,\, i,\, l}\) in (2), and the channel capacity becomes:$$\begin{aligned} C&= \frac{L_{d2}}{2NL_{d2}} \nonumber \\&= \frac{1}{2N} \qquad \text{ bits/transmission } \end{aligned}$$(20)per dimension. Substituting (20) and (21) in (18), and using (19) we get the minimum (threshold) average SNR per bit required for error-free transmission, for a given channel capacity. The minimum SNR per bit for various code rates and receiver diversity is presented in Table 2. Note that [50] the minimum \(E_b/N_0\) for error-free transmission is \(-1.6\) dB only when \(C\rightarrow 0\).$$\begin{aligned} C&= \frac{L_{d2}}{4NL_{d2}} \nonumber \\&= \frac{1}{4N} \qquad \text{ bits/transmission } \end{aligned}$$(21)
- 10.In the case of fading channels, it may not be possible to achieve the minimum possible SNR per bit. This is because, the SNR per bit of a given frame may be less than the threshold average SNR per bit. Such frames are said to be in outage. The frame SNR per bit can be defined as (for the \(k\)th frame and the \(l\)th diversity arm):where \(\langle \cdot \rangle \) denotes time average over the \(L_{d2}\) data symbols. Note that the frame SNR is different from the average SNR per bit, which is defined as [4]:$$\begin{aligned} \text{ SNR }_{k,\, l,\,\mathrm {bit}} = \frac{1}{2C} \frac{\langle |\tilde{H}_{k,\, i,\, l} S_{k,\, 3,\, i}|^2\rangle }{\langle |\tilde{W}_{k,\, i,\, l}|^2\rangle } \end{aligned}$$(22)The \(k\)th OFDM frame is said to be in outage when:$$\begin{aligned} \text{ SNR } \text{ per } \text{ bit }&= \frac{1}{2C} \frac{E\left[ \left| \tilde{H}_{k,\, i,\, l} S_{k,\, 3,\, i} \right| ^2\right] }{E\left[ \left| \tilde{W}_{k,\, i,\, l}\right| ^2\right] } \nonumber \\&= \frac{1}{2C} \frac{2 L_h \sigma ^2_f}{L_d \sigma ^2_w}. \end{aligned}$$(23)for all \(l\). The outage probability is given by:$$\begin{aligned} \text{ SNR }_{k,\, l,\,\mathrm {bit}} < \text{ minimum } \text{ average } \text{ SNR } \text{ per } \text{ bit } \end{aligned}$$(24)$$\begin{aligned} P_{\mathrm {out}} = \frac{\text{ number } \text{ of } \text{ frames } \text{ in } \text{ outage }}{\text{ total } \text{ number } \text{ of } \text{ frames } \text{ transmitted }}. \end{aligned}$$(25)

The minimum SNR per bit for different code rates and receiver diversity

Rate-1/2 turbo code \(1\)st-order receive diversity | Rate-1 turbo code \(1\)st-order receive diversity | Rate-1 turbo code 2nd-order receive diversity | |
---|---|---|---|

Min avg SNR per bit (dB) | \(-\)0.817 | 0 | \(-\)0.817 |

## 6 Simulation Results

Similarly we observe from Fig. 11 that, with 2nd order receive diversity, the outage probability is \(10^{-4}\) at 3 dB SNR per bit. This implies that 1 out of \(10^{4}\) frames is in outage. Using similar arguments, the best case overall BER at 3 dB SNR per bit would be \(0.5\times 3{,}832/(10{,}000\times 3{,}832)=0.5\times 10^{-4}\). From Fig. 10, the ideal coherent receiver gives a BER of \(2\times 10^{-2}\), at 3 dB SNR per bit, once again suggesting that there is large scope for improvement.

## 7 Conclusions and Future Work

This paper deals with linear complexity coherent detectors for turbo-coded OFDM signals transmitted over frequency selective Rayleigh fading channels. Simulation results show that it is possible to achieve a BER of \(10^{-5}\) at an SNR per bit of 8 dB and throughput equal to 82.84 %, using a single transmit and two receive antennas.

With the rapid advances in VLSI technology, it is expected that coherent transceivers would drive the future wireless telecommunication systems.

It may be possible to further improve the performance, using a better code.

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